We use a weighted analogue of a trace to define a weighted noncommutative decreasing rearrangement and show it's relationship with the singular value function. We further show an alternative approach to constructing weighted non-commutative Banach function spaces using weighted non-commutative decreasing rearrangements and prove that this approach is equivalent to the original approach by Labuschagne and Majewski.

Samevatting iv Sleutelwoorde iv 1.3. Noncommutative Banach function spaces 9 Chapter 2. Weighted Banach Function Spaces 2.1. A first definition of weighted non-commutative Banach function spaces 2.2. The map τ x 2.3. Weighted non-commutative decreasing rearrangements 2.4. Equivalence of weighted spaces Chapter 3. Weighted Orlicz Spaces 3.1. Young functions and Orlicz spaces 3.2. Equivalence of weighted Orlicz spaces 3.3. The Köthe dual of weighted noncommutative Orlicz spaces Chapter 4. Interpolation Spaces of Weighted Banach Function Spaces

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